Question 1140751: Bonus: The number 15 is the smallest positive number which is a sum of two consecutive numbers, three consecutive numbers, and five consecutive numbers:
15 = 7+8
15 = 4+5+6
15 = 1+2+3+4+5
What is the largest number under 1000 that satisfies the same conditions?
Answer by greenestamps(13200)   (Show Source):
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(1) The sum of 3 consecutive numbers is 3 times the number in the middle; the sum of 5 consecutive numbers is 5 times the number in the middle. So any number that can be written as the sum of both 3 consecutive numbers and 5 consecutive numbers must be a multiple of both 3 and 5, and therefore must be a multiple of 15.
(2) The sum of 2 consecutive numbers is always odd.
(3) From (1) and (2), the numbers that can be written as the sum of 2, 3, or 5 consecutive numbers are all the odd multiples of 15.
(4) The largest odd multiple of 15 less than 1000 is 975.
ANSWER: 975
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